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Rational Functions (College Board AP® Precalculus): Exam Questions

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Rational Functions

1
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1 mark

The function space f is given by space f open parentheses x close parentheses equals fraction numerator 3 x minus 2 over denominator x squared plus 4 x minus 5 end fraction . Which of the following statements about the end behavior of space f is true?

  • The graph of space f has a horizontal asymptote at space y equals 0 because the polynomial in the denominator dominates the polynomial in the numerator for input values of large magnitude.

  • The graph of space f has a horizontal asymptote at space y equals 3 because the quotient of the leading terms gives 3.

  • The graph of space f has a slant asymptote because the polynomial in the denominator has a greater degree than the polynomial in the numerator.

  • The graph of space f has no horizontal asymptote because neither polynomial dominates the other for input values of large magnitude.

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2
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1 mark

The rational function space f open parentheses x close parentheses equals fraction numerator 2 x squared plus 5 x minus 3 over denominator x squared minus 9 end fraction can be rewritten in the equivalent factored form space f open parentheses x close parentheses equals fraction numerator open parentheses 2 x minus 1 close parentheses open parentheses x plus 3 close parentheses over denominator open parentheses x minus 3 close parentheses open parentheses x plus 3 close parentheses end fraction . Which of the following features of the graph of space f is most readily revealed by this factored form?

  • The graph has a vertical asymptote at x equals negative 3.

  • The graph has a hole at x equals negative 3.

  • The graph has a vertical asymptote at space x equals 2.

  • The graph has an x-intercept at x equals 3.

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3
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1 mark

A manufacturer determines that the total cost, in dollars, to produce x units of a product is given by C open parentheses x close parentheses equals 500 plus 12 x. The average cost per unit is modeled by the rational function A open parentheses x close parentheses equals fraction numerator 500 plus 12 x over denominator x end fraction. What is the value of A open parentheses 50 close parentheses and what does it represent in this context?

  • A \left(50\right) = 22; the average cost per unit is $22 when 50 units are produced.

  • A \left(50\right) = 22; the total cost to produce 50 units is $22.

  • A \left(50\right) = 1 , 100; the average cost per unit is $1,100 when 50 units are produced.

  • A \left(50\right) = 1 , 100; the total cost to produce 50 units is $1,100.

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4
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1 mark

The function r is given by r open parentheses x close parentheses equals fraction numerator x squared minus 16 over denominator x squared plus 7 x plus 12 end fraction.

At what value of x does the graph of r have a hole?

  • x equals negative 4

  • x equals negative 3

  • x equals 3

  • x equals 4

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5
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1 mark

The function r is given by r open parentheses x close parentheses equals fraction numerator open parentheses x minus 2 close parentheses open parentheses x minus 3 close parentheses over denominator x plus 1 end fraction.

What are all real zeros of r?

  • x equals negative 3 and x equals negative 2

  • x equals negative 1

  • x equals negative 1, x equals 2, and x equals 3

  • x equals 2 and x equals 3

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6
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1 mark

In the x y-plane, the graph of a rational function f has a vertical asymptote at x = 3. Which of the following could be an expression for f \left(x\right) ?

  • \frac{\left(x - 3\right) \left(x + 1\right)}{2 \left(x + 3\right)}

  • \frac{\left(x + 2\right) \left(x - 3\right)}{\left(x - 1\right) \left(x - 3\right)}

  • \frac{\left(x - 3\right) \left(x + 4\right)}{\left(x + 3\right) \left(x - 1\right)}

  • \frac{\left(x + 3\right) \left(x + 1\right)}{\left(x + 1\right) \left(x - 3\right)}

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7
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1 mark

The rational function r is given by r open parentheses x close parentheses equals fraction numerator 4 x squared minus 3 x plus 1 over denominator 2 x squared plus 5 end fraction . Which of the following describes the end behavior of r as x decreases without bound?

  • limit as x rightwards arrow negative infinity of r open parentheses x close parentheses equals infinity

  • limit as x rightwards arrow negative infinity of r open parentheses x close parentheses equals 2

  • limit as x rightwards arrow negative infinity of r open parentheses x close parentheses equals 0

  • limit as x rightwards arrow negative infinity of r open parentheses x close parentheses equals negative infinity

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8
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1 mark

Which of the following functions has a zero at x = -2 and has a graph in the xy-plane with a vertical asymptote at x = 3 and a hole at x = -1 ?

  • h \left(x\right) = \dfrac{x^{2} + 3x + 2}{x^{2} - 2x - 3}

  • j \left(x\right) = \dfrac{x^{2} + x - 2}{x^{2} - 2x - 3}

  • k \left(x\right) = \dfrac{x + 2}{x^{2} - 2x - 3}

  • m \left(x\right) = \dfrac{x^{2} + 3x + 2}{x^{2} + 2x - 3}

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9
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1 mark

The function g is given by g(x) = x^3 - x^2 - 6x, and the function h is given by h(x) = x^2 + 4x - 5. Let k be the function given by k left parenthesis x right parenthesis equals fraction numerator h left parenthesis x right parenthesis over denominator g left parenthesis x right parenthesis end fraction. What is the domain of k?

  • All real numbers x where x \neq 0

  • All real numbers x where x \neq -5 and x \neq 1

  • All real numbers x where x \neq -2, x \neq 0, and x \neq 3

  • All real numbers x where x \neq -5, x \neq -2, x \neq 0, x \neq 1, and x \neq 3

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10
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1 mark

The rational function g is given by g open parentheses x close parentheses equals fraction numerator x plus 2 over denominator open parentheses x minus 3 close parentheses squared end fraction . Which of the following correctly describes the unbounded behavior of g near x equals 3 ?

  • limit as x rightwards arrow 3 to the power of minus of g open parentheses x close parentheses equals infinity and limit as x rightwards arrow 3 to the power of plus of g open parentheses x close parentheses equals infinity

  • limit as x rightwards arrow 3 to the power of minus of g open parentheses x close parentheses equals negative infinity and limit as x rightwards arrow 3 to the power of plus of g open parentheses x close parentheses equals infinity

  • limit as x rightwards arrow 3 to the power of minus of g open parentheses x close parentheses equals infinity and limit as x rightwards arrow 3 to the power of plus of g open parentheses x close parentheses equals negative infinity

  • limit as x rightwards arrow 3 to the power of minus of g open parentheses x close parentheses equals negative infinity and limit as x rightwards arrow 3 to the power of plus of g open parentheses x close parentheses equals negative infinity

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11
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1 mark

The graph of the rational function h open parentheses x close parentheses equals fraction numerator x squared plus 2 x minus 8 over denominator x minus 2 end fraction has a hole at x equals 2. What are the coordinates of the hole?

  • \left(2 , 0\right)

  • \left(2 , 4\right)

  • \left(2 , 6\right)

  • \left(2 , \infty\right)

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12
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1 mark

The function r is given by r open parentheses x close parentheses equals fraction numerator x plus 5 over denominator x squared minus 9 x plus 18 end fraction.

What is the domain of r?

  • all real numbers x

  • all real numbers x where x not equal to negative 5

  • all real numbers x where x not equal to 3 and x not equal to 6

  • all real numbers x where x \neq - 5, x not equal to 3, and x \neq 6

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13
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1 mark

The function r is given by r open parentheses x close parentheses equals fraction numerator open parentheses x minus 1 close parentheses squared open parentheses x plus 3 close parentheses over denominator open parentheses x minus 1 close parentheses open parentheses x plus 2 close parentheses end fraction.

The graph of r in the x y-plane has one hole. Which of the following are the coordinates of the hole in the graph of r?

  • \left(- 3 , 0\right)

  • \left(- 2 , 0\right)

  • \left(1 , 0\right)

  • \left(1 , \frac{4}{3}\right)

Choose your answer

14
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1 mark

The function space p is given by space p open parentheses x close parentheses equals fraction numerator 2 x squared plus 5 x minus 3 over denominator x plus 1 end fraction.

Which of the following is the equation of the slant asymptote of the graph of space p?

  • y = 2 x - 3

  • y = 2 x + 3

  • y = 2 x + 5

  • y = 2 x + 6

Choose your answer

15
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1 mark

Let space p open parentheses x close parentheses equals fraction numerator x minus 4 over denominator x plus 1 end fraction . On which of the following intervals is space p open parentheses x close parentheses greater or equal than 0 ?

  • \left[- 1 , 4\right]

  • \left(- 1 , 4\right]

  • \left(- \infty , - 1\right) \cup \left[4 , \infty\right)

  • \left(- \infty , - 1\right] \cup \left[4 , \infty\right)

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16
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1 mark

Let f be a rational function that is graphed in the x y-plane. Consider x = 2 and x = - 4. The polynomial in the numerator of f has a zero at x = 2 and does not have a zero at x = - 4. The polynomial in the denominator of f has zeros at both x = 2 and x = - 4. The multiplicities of the zeros at x = 2 in the numerator and in the denominator are equal. Which of the following statements is true?

  • The graph of f has holes at both x = 2 and x = - 4.

  • The graph of f has a vertical asymptote at x = 2 and a hole at x = - 4.

  • The graph of f has a hole at x = 2 and a vertical asymptote at x = - 4.

  • The graph of f has vertical asymptotes at both x = 2 and x = - 4.

Choose your answer

17
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1 mark

The function m is given by m open parentheses x close parentheses equals fraction numerator 2 x squared minus 5 x plus 1 over denominator x minus 3 end fraction . The graph of m has a slant asymptote. Which of the following is the equation of the slant asymptote?

  • y = 2 x - 11

  • y = 2 x - 5

  • y = 2 x + 1

  • y = x - 3

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